Classical position probability distribution in stationary and separable quantum systems

Despite various parallels between quantum states and ordinary information, quantum no-go-theorems have convinced many that there is no realistic framework that might underly quantum theory, no reality that quantum states can represent knowledge about. This paper develops the case that there is a plausible underlying reality: one actual spacetime-based history, although with behavior that appears strange when analyzed dynamically (one time-slice at a time). By using a simple model with no dynamical laws, it becomes evident that this behavior is actually quite natural when analyzed “all-at-once” (as in classical action principles). From this perspective, traditional quantum states would represent incomplete information about possible spacetime histories, conditional on the future measurement geometry. Without dynamical laws imposing additional restrictions, those histories can have a classical probability distribution, where exactly one history can be said to represent an underlying reality.

We study the extent to which ψ-epistemic models for quantum measurement statistics-models where the quantum state does not have a real, ontic status-can explain the indistinguishability of nonorthogonal quantum states. This is done by comparing the overlap of any two quantum states with the overlap of the corresponding classical probability distributions over ontic states in a ψ-epistemic model. It is shown that in Hilbert spaces of dimension d≥4, the ratio between the classical and quantum overlaps in any ψ-epistemic model must be arbitrarily small for certain nonorthogonal states, suggesting that such models are arbitrarily bad at explaining the indistinguishability of quantum states. For dimensions d=3 and 4, we construct explicit states and measurements that can be used experimentally to put stringent bounds on the ratio of classical-to-quantum overlaps in ψ-epistemic models, allowing one in particular to rule out maximally ψ-epistemic models more efficiently than previously proposed.

Fermions from classical statistics

We demonstrate that the Wigner function of a pure quantum state is a wave\nfunction in a specially tuned Dirac bra-ket formalism and argue that the Wigner\nfunction is in fact a probability amplitude for the quantum particle to be at a\ncertain point of the classical phase space. Additionally, we establish that in\nthe classical limit, the Wigner function transforms into a classical\nKoopman-von Neumann wave function rather than into a classical probability\ndistribution. Since probability amplitude need not be positive, our findings\nprovide an alternative outlook on the Wigner function's negativity.\n

In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, equations for the Green function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green function of the Schr\"odinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the definition recovering well known results. As an application, the forced parametric oscillator is considered. Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.

We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.

The deficiencies existing in Geotechnical parameters distribution fitting using the classical probability distribution are analyzed and the stochastic weighting method is employed to improve the the small samples of Geotechnical parameters into big samples, thus sloving the problems caused by the small sample. The maximum entropy principle and the big samples improving from small sample are used to generate probability density functions of Geotechnical parameters. It is more rational and scientific with the method proposed in the paper through the Kolmogorov-Smirnov Test. It is demonstrated through the example that the method presented in this paper not only successfully escapes the reliance of the classical probability distribution functions, but also make the computation result more closed to the fact due to the data coming from the big sample improving from the small sample, which has important engineering significance.

We review the methods and use of random quantum states with particular emphasis on recent theoretical developments and applications in various fields. The guiding principle of the review is the idea that random quantum states can be understood as classical probability distributions in the Hilbert space of the associated quantum system. We show how this central concept connects questions of physical interest that cover different fields such as quantum statistical physics, quantum chaos, mesoscopic systems of both non-interacting and interacting particles, including superconducting and spin–orbit phenomena, and stochastic Schrodinger equations describing open quantum systems.

An efficient quantum secret sharing scheme is proposed, in which the quantum state and the classical information can be shared simultaneously through only one distribution. The dealer uses the operations of quantum-controlled-not and Hadamard gate to encode the secret quantum state and classical information, and the participants use the single-particle measurements to recover the original quantum state and classical information. Compared to the existing schemes, our scheme is more efficient when the quantum state and the classical information need to be shared simultaneously.

When a quantum system is divided into two local subsystems, measurements on the two subsystems can exhibit correlations beyond those possible in a classical joint probability distribution; these are partially explained by entanglement, and more generally by a wider class of measures such as the quantum discord. In this paper, I introduce a simple thought experiment defining a measure of quantum correlations, which I call the accord, and write the result as a minimax optimization over unitary matrices. I find the exact result for pure states as a simple function of the Schmidt coefficients and provide a complete proof, and I likewise provide and prove the result for several classes of mixed states, notably including all states of two qubits and the experimentally relevant case of a pure state mixed with colorless noise. I demonstrate that for two qubit states the accord provides a tight lower bound on the discord; for Bell diagonal states it is also an upper bound on entanglement.

Analytical results for the classical and semiclassical coordinate probability distribution function of anharmonic oscillators

Predicting and modeling sports events has always been one of the main applications of probability and statistics. From analyzing the likelihood of a team's victory based on historical performance data can be used to estimate the probability of a particular player scoring a goal. This paper focuses on the practical use of probability in football matches, summarizing several innovative prediction models based on classical probability distributions. It demonstrates the accuracy of the bivariate Poisson distribution model in predicting football scores, especially the number of draws through examples. And visually illustrate the precise capture of goals and winning moments in football matches by the In-game win probability model which based on Bayesian statistical framework. The paper also introduces the broad integration prospects of probability in combination with other disciplines in sports events. These interdisciplinary approaches have significant implications for not only helps optimize team organization and management like tactical improvement and risk control, but also plays a key role in promoting the development of related industries such as sports betting and media broadcasting.

The dynamics of quantum and classical probability distributions are computed for a model of two coupled rotors (or pendulums), with emphasis on the $\ensuremath{\hbar}$ scaling of the quantum-classical (QC) differences. This scaling is not the same for coarse-grained and fine-grained quantities. The QC differences in the averages of observables scale as ${\ensuremath{\hbar}}^{2∕3}$ for chaotic states, but as ${\ensuremath{\hbar}}^{2}$ for regular states. The QC differences in probability distributions scale as ${\ensuremath{\hbar}}^{1∕3}$ for chaotic states. No simple scaling is found for those differences in regular states, although their overall magnitudes are similar to those in chaotic states. QC differences arise first in the short-wavelength regime, and subsequently spread to all wavelengths. A satisfactory classical limit is obtained without invoking environmental decoherence, which in any case would be effective in suppressing only short-wavelength QC differences.

Due to various uncertainties, most of geotechnical parameters are small samples, which causes much trouble when the probability distribution of geotechnical parameter is fitted using traditional distributions. This paper uses stochastic weighted method to improve the small samples of Geotechnical parameters into big samples, thus solving the problems caused by the small samples. Meanwhile, the probability density function of geotechnical parameter is derived based on maximum entropy principle, the advantage of presented method is verified through Kolmogorov-Smirnov Test. Case study shows that the proposed method not only overcomes the dependence of conventional fitting methods on classical probability distributions, but also the fitting more close to the fact because the data come from the big sample improved by geotechnical parameters, which has important engineering significance.

Quantum channels are trace-preserving completely positive linear maps between Banach spaces of trace-class operators (Schatten classes of order 1); these are noncommutative analogs of Markov operators in classical probability theory. They also play the role of dynamical maps in quantum theory [1, Chap. 6]. The main characteristics determining the information properties of a quantum channel include its classical entanglement-assisted and unassisted capacities. The classical (unassisted) capacity C(Φ) of a channel Φ determines the limit rate of classical information transmission through Φ with any block coding at the input and the corresponding measurement at the output, and the classical entanglement-assisted channel capacity Cea(Φ) supposes, in addition, the presence of an entangled state between the input and the output of the channel Φ (a detailed description of transmission protocols can be found in [1, Chap. 8]). Since entanglement is an additional resource, it follows that Cea(Φ) ≥ C(Φ) for any channel Φ. LetH be a separable Hilbert space. By T(H) we denote the Banach space of all trace-class operators onH and byS(H), the subset of T(H) consisting of all positive operators with trace 1; we refer to such operators as quantum states and denote them by Greek letters ρ, σ, . . . . We denote a set of quantum states {ρi} with probability distributions {πi} by {πi, ρi} and call it an ensemble of states; the state ρ = ∑ i πiρi is called the average state of the ensemble {πi, ρi}. A quantum channel is a trace-preserving completely positive linear map Φ: T(HA) → T(HB) [1, Chap. 6]. Let H(ρ) be the von Neumann entropy of a state ρ, and let H(ρ‖σ) be the quantum relative entropy of states ρ and σ. For a given channel Φ and any ensemble {πi, ρi} of input quantum states, the output χ-quantity is determined by the expression